On the joint observability of flow fields and particle properties from Lagrangian trajectories: evidence from neural data assimilation

We numerically investigate the joint observability of flow states and unknown particle properties from Lagrangian particle tracking (LPT) data. LPT offers time-resolved, volumetric measurements of particle trajectories, but experimental tracks are spatially sparse, potentially noisy, and may be further complicated by inertial transport, raising the question of whether both Eulerian fields and particle characteristics can be reliably inferred. To address this, we develop a data assimilation framework that couples an Eulerian flow representation with Lagrangian particle models, enabling the simultaneous inference of carrier fields and particle properties under the governing equations of disperse multiphase flow. Using this approach, we establish empirical existence proofs of joint observability across three representative regimes. In a turbulent boundary layer with noisy tracer tracks (St to 0), flow states and true particle positions are jointly observable. In homogeneous isotropic turbulence seeded with inertial particles (St ~ 1-5), we demonstrate simultaneous recovery of flow states and particle diameters, showing the feasibility of implicit particle characterization. In a compressible, shock-dominated flow, we report the first joint reconstructions of velocity, pressure, density, and inertial particle properties (diameter and density), highlighting both the potential and certain limits of observability in supersonic regimes. Systematic sensitivity studies further reveal how seeding density, noise level, and Stokes number govern reconstruction accuracy, yielding practical guidelines for experimental design. Taken together, these results show that the scope of LPT could be broadened to multiphase and high-speed flows, in which tracer and measurement fidelity are limited.